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A note on nonautonomous logistic equation with random perturbation. (English) Zbl 1076.34062

Consider the nonautonomous extension of randomized logistic equation

dN(t)=N(t)[(a(t)-b(t)N(t))dt+α(t)dB(t)],N(0)=N 0 >0,t0,

driven by a 1-dimensional Brownian motion B on a probability space (Ω,,( t ) t0 ,P). Suppose that the coefficients a,b,α are continuous, T-periodic functions satisfying

a(t)>0,b(t)>0, 0 T [a(s)-α 2 (s)]ds>0·

This note shows that E[1/N(t)] has a unique positive T-periodic solution under these conditions as a natural extension of a well-known property of the underlying deterministic model.

34F05ODE with randomness
60H10Stochastic ordinary differential equations
37H10Generation, random and stochastic difference and differential equations
60H30Applications of stochastic analysis
92D25Population dynamics (general)